We know C=F, -40. What is |K|=|F|?

Originally posted by SwissGambit
I already had found that easier path. See post 4 on page 1:

(x-273.15)*1.8+32=-x
1.8x -491.67+32=-x
-459.67=-2.8x
x=164.1678571

Go back to:

-459.67=-2.8x

Multiply both sides by 100:

-45967=-280x
x=45967/280

But I was showing sonhouse a general method for converting ANY number with a repeating decimal to a fraction. I just happened to use that number as an example.

I really appreciate the effort, guys. I have some practicing to do to master this technique. Thanks again.

Originally posted by sonhouse
Well my calculator says 45967/280 covers the 164.1678571 part exactly but it misses out on the actual repeaters, the 428571's that repeat forever.

How did you suss out that fraction anyway? You have fraction sniffing software?

When you see something in an equation in the form of a decimal just convert it to it's fractional equivalent (do this for all terms). Then just simplify and solve.

i.e. (K-273.15)*1.8 +32 = +/-K becomes
((100K - 27315)/100)(9/5) + 32 = +/-K


==> 9(20K - 5463) +3200 = +/-100K
==> 180 +/- 100K = 49167 - 3200
==> K = 45967/(180 +/- 100)


Also, when you see a recuring decimal, you can quickly convert to its exact fractional form, by expressing it as a sum of the repeating and non repeating parts (where the repeating part can be written as x/D(x) where x is the repeated term, and D(x) is 10^(number of digits in x) -1. i.e. with x = 123 then 0.123123123123 = 123/(D(123)) = 123/999
This works because if (relabeling x)
[1] x = f + bar(y) (f is some fraction, y a repeated term in the decimal part) then
(10^d)x = (10^d)f + y + bar(y) (d is the number of digits in y)


==> (10^d - 1)x = (10^d - 1)f + y (subtracting [1])
==> x = f +y/(10^d - 1)

Originally posted by Agerg
When you see something in an equation in the form of a decimal just convert it to it's fractional equivalent (do this for all terms). Then just simplify and solve.

i.e. (K-273.15)*1.8 +32 = +/-K becomes
((100K - 27315)/100)(9/5) + 32 = +/-K


==> 9(20K - 5463) +3200 = +/-100K
==> 180 +/- 100K = 49167 - 3200
==> K = 45967/(180 +/- 100)

Ugh, all that extra clutter in the intermediate steps.

(x-273.15)*1.8+32=+/-x
1.8x-491.67+32=+/-x
-459.67=-1.8+/-1(x)

And the fraction emerges naturally:

x = 45967/(180 +/- 100)

Originally posted by SwissGambit
Ugh, all that extra clutter in the intermediate steps.

(x-273.15)*1.8+32=+/-x
1.8x-491.67+32=+/-x
-459.67=-1.8+/-1(x)

And the fraction emerges naturally:

x = 45967/(180 +/- 100)

Ok...I think it's a matter of preference though - I prefer to clear out decimals as early as possible.

Originally posted by Agerg
Also, when you see a recuring decimal, you can quickly convert to its exact fractional form, by expressing it as a sum of the repeating and non repeating parts (where the repeating part can be written as x/D(x) where x is the repeated term, and D(x) is 10^(number of digits in x) -1. i.e. with x = 123 then 0.123123123123 = 123/(D(123)) = 123/999
This works bec ...[text shortened]... f digits in y)


==> (10^d - 1)x = (10^d - 1)f + y (subtracting [1])
==> x = f +y/(10^d - 1)

How does this method handle a number like 0.000 123 123 123 123 123... ?

Originally posted by SwissGambit
How does this method handle a number like 0.000 123 123 123 123 123... ?

just multiply by enough "0s" to left shift by the length of 1 repeating part:
x = 0.000 123 123 123 123 123...

1000x = 0.123 123 123 123...

999x = 0.123

999x = 123/1000

x = 123/999000

x = (3*41)/(3*333000)

x = 41/333000

If you like recurring decimals, continued fractions are really good too, they turn any root of any number into a recurring fraction.

Originally posted by SwissGambit
How does this method handle a number like 0.000 123 123 123 123 123... ?

hmm...good point, I retract the method and "proof" - if there is non-recurring decimal part, account of this (in particular the length) needs to be made when looking the recurring part (as iamatiger suggests)

Damn that's embarrassing! 😞

Here's another one to try.

136.6064 3564 3564 3564...

What is the simplest fraction?

I don't see how a preference for fractions over decimals helps on this one.

Originally posted by Agerg
hmm...good point, I retract the method and "proof" - if there is non-recurring decimal part, account of this (in particular the length) needs to be made when looking the recurring part (as iamatiger suggests)

Damn that's embarrassing! 😞

Nah, no need to retract - just modify the method as iamatiger did, and it's fine.